This hasn't been my experience. I worked with a man years ago that was relatively fluent in English, but whose thick accent made understanding him somewhat difficult. Listening intently was required to understand him and he would often need to repeat himself, even though he was using the correct words and grammar. It's something he ran into pretty often at the time and he worked to improve it. Something like this app may have helped him. Slight accents may be "just top dressing," but this is most definitely not always the case.